RMS value of alternating currents and voltages. Voltage rms value

The alternating sinusoidal current has different instantaneous values ​​during the period. It is natural to ask the question, what value of the current will be measured by the ammeter included in the circuit?

When calculating alternating current circuits, as well as electrical measurements, it is inconvenient to use instantaneous or amplitude values ​​of currents and voltages, and their average values ​​over a period are zero. In addition, the electrical effect of a periodically varying current (the amount of heat released, the perfect work, etc.) cannot be judged by the amplitude of this current.

The most convenient was the introduction of the concepts of the so-called effective values ​​of current and voltage... These concepts are based on the thermal (or mechanical) action of the current, which does not depend on its direction.

This is the value of direct current at which the same amount of heat is generated in the conductor during the period of alternating current as during alternating current.

To evaluate the action produced, we compare its action with the thermal effect of direct current.

The power P of direct current I passing through the resistance r will be P = P 2 r.

AC power will be expressed as the average effect of the instantaneous power I 2 r over the whole period or the average value of (Im x sinω t) 2 x r in the same time.

Let the average value of t2 over the period be M. Equating the DC power and AC power, we have: I 2 r = Mr, whence I = √ M,

The magnitude I is called the rms value of the alternating current.

The average value of i2 at alternating current is determined as follows.

Let's build a sinusoidal current curve. Squaring each instantaneous current value, we get a curve of P versus time.

Both halves of this curve lie above the horizontal axis, since negative current values ​​(-i) in the second half of the period, when squared, give positive values.

Let's construct a rectangle with base T and area equal to the area bounded by the curve i 2 and the horizontal axis. The height of the rectangle M will correspond to the average value of P over the period. This value for the period, calculated using higher mathematics, will be equal to 1 / 2I 2 m. Therefore, М = 1 / 2I 2 m

Since the rms value of the alternating current is equal to I = √ M, then finally I = Im / √ 2

Similarly, the relationship between the effective and amplitude values ​​for the voltage U and E has the form:

U = Um / √ 2 E = Em / √ 2

The effective values ​​of variables are indicated by capital letters without subscripts (I, U, E).

Based on the above, we can say that the effective value of the alternating current is equal to that direct current, which, passing through the same resistance as the alternating current, releases the same amount of energy at the same time.

Electrical measuring instruments (ammeters, voltmeters) connected to the alternating current circuit show the effective values ​​of current or voltage.

When constructing vector diagrams, it is more convenient to postpone not the amplitude, but the effective values ​​of the vectors. For this, the lengths of the vectors are reduced by √ 2 times. This does not change the location of the vectors on the diagram.

Values ​​of effective voltage and current. Definition. Amplitude ratio for different shapes. (10+)

The concept of effective (effective) values ​​of voltage and current

When we talk about alternating voltages or currents, especially of a complex shape, the question arises of how to measure them. After all, the tension is constantly changing. You can measure the signal amplitude, that is, the maximum modulus of the voltage value. This measurement method is fine for relatively smooth signals, but the presence of short bursts spoils the picture. Another criterion for choosing a measurement method is for what purpose the measurement is made. Since in most cases the power that a particular signal can give is of interest, the effective (effective) value is used.

,

After substitution of the current value i and subsequent transformations, we get that the effective value of the alternating current is:

Similar ratios can also be obtained for voltage and EMF:

Most electrical measuring instruments measure not instantaneous, but effective values ​​of currents and voltages.

Considering, for example, that the effective voltage value in our network is 220V, it is possible to determine the amplitude value of the voltage in the network: U m = UÖ2 = 311V. It is important to consider the relationship between the effective and peak values ​​of voltages and currents, for example, when designing devices using semiconductor elements.

R.m.s. value of alternating current

Theory/ TOE/ Lecture N 3. Representation of sinusoidal values ​​using vectors and complex numbers.

Alternating current has not found practical use for a long time. This was due to the fact that the first generators of electrical energy produced direct current, which fully satisfied the technological processes of electrochemistry, and direct current motors have good control characteristics. However, with the development of production, direct current became less and less suitable for the increasing requirements of an economical power supply. Alternating current made it possible to effectively crush electrical energy and change the voltage value using transformers. The possibility of generating electricity at large power plants with its subsequent economical distribution to consumers has appeared, the radius of power supply has increased.

Currently, the central production and distribution of electrical energy is carried out mainly on alternating current. Circuits with changing - alternating - currents in comparison with direct current circuits have a number of features. Alternating currents and voltages cause alternating electric and magnetic fields. As a result of changes in these fields in the circuits, the phenomena of self-induction and mutual induction arise, which have the most significant effect on the processes occurring in the circuits, complicating their analysis.

Alternating current (voltage, emf, etc.) is a current (voltage, emf, etc.) that changes over time. Currents whose values ​​are repeated at regular intervals in the same sequence are called periodic, and the smallest time interval after which these repetitions are observed is period T. For a periodic current, we have

The frequency range used in technology: from ultra-low frequencies (0.01¸10 Hz - in automatic control systems, in analog computers) - to ultrahigh (3000 ¸ 300000 MHz - millimeter waves: radar, radio astronomy). RF industrial frequency f= 50Hz.

The instantaneous value of a variable is a function of time. It is customary to denote it with a lowercase letter:

i- instantaneous current value;

u- instantaneous voltage value;

e- instantaneous value of EMF;

R- instantaneous power value.

The largest instantaneous value of a variable over a period is called amplitude (it is customary to denote it by a capital letter with an index m).

Amplitude of the current;

Voltage amplitude;

EMF amplitude.

The value of a periodic current equal to such a value of a direct current that, during one period, will produce the same thermal or electrodynamic effect as a periodic current is called effective value periodic current:

The effective values ​​of EMF and voltage are determined in a similar way.

Sinusoidally varying current

Of all the possible forms of periodic currents, the sinusoidal current is the most widespread. Compared with other types of current, sinusoidal current has the advantage that in the general case it allows the most economical production, transmission, distribution and use of electrical energy. Only when using a sinusoidal current is it possible to keep the shape of the curves of voltages and currents unchanged in all sections of a complex linear circuit. Sinusoidal current theory is the key to understanding other circuit theory.

Image of sinusoidal emf, voltages and currents on the plane of Cartesian coordinates

Sinusoidal currents and voltages can be displayed graphically, written using equations with trigonometric functions, represented as vectors on the Cartesian plane or complex numbers.

Shown in Fig. 1, 2 graphs of two sinusoidal EMF e 1 and e 2 correspond to the equations:

The values ​​of the arguments of sinusoidal functions are called phases sinusoids, and the phase value at the initial moment of time (t=0): and - initial phase ( ).

The quantity characterizing the rate of change of the phase angle is called angular frequency. Since the phase angle of the sinusoid during one period T changes to rad., then the angular frequency is , where f– frequency.

When two sinusoidal quantities of the same frequency are considered together, the difference in their phase angles, equal to the difference in the initial phases, is called phase angle.

For sinusoidal EMF e 1 and e 2 phase angle:

Vector image of sinusoidally varying quantities

On the Cartesian plane from the origin, vectors are drawn that are equal in magnitude to the amplitude values ​​of sinusoidal quantities, and these vectors are rotated counterclockwise ( in TOE this direction is taken as positive) with an angular frequency equal to w... The phase angle during rotation is measured from the positive abscissa semiaxis. The projections of the rotating vectors on the ordinate axis are equal to the instantaneous values ​​of the EMF e 1 and e 2 (fig. 3). The set of vectors depicting sinusoidally changing EMF, voltages and currents is called vector diagrams. When constructing vector diagrams, it is convenient to place vectors for the initial moment of time (t=0), which follows from the equality of the angular frequencies of the sinusoidal quantities and is equivalent to the fact that the Cartesian coordinate system itself rotates counterclockwise with the speed w... Thus, in this coordinate system the vectors are motionless (Fig. 4). Vector diagrams are widely used in the analysis of sinusoidal current circuits. Their use makes the calculation of the chain more visual and simple. This simplification is that the addition and subtraction of the instantaneous values ​​of the quantities can be replaced by the addition and subtraction of the corresponding vectors.

Let, for example, at the branching point of the circuit (Fig. 5), the total current is equal to the sum of the currents and two branches:

Each of these currents is sinusoidal and can be represented by the equation

The resulting current will also be sinusoidal:

Determination of the amplitude and the initial phase of this current by means of appropriate trigonometric transformations turns out to be rather cumbersome and not very clear, especially if a large number of sinusoidal quantities are summed up. It is much easier to do this using a vector diagram. In fig. 6 shows the initial positions of the current vectors, the projections of which on the ordinate axis give the instantaneous values ​​of the currents for t=0. When these vectors rotate with the same angular velocity w their relative position does not change, and the phase angle between them remains equal.

Since the algebraic sum of the vector projections on the ordinate axis is equal to the instantaneous value of the total current, the total current vector is equal to the geometric sum of the current vectors:

.

The construction of a vector diagram on a scale allows you to determine the values ​​and from the diagram, after which the solution for the instantaneous value can be written by formally taking into account the angular frequency:.

RMS and average values ​​of alternating current and voltage.

Average or arithmetic mean Fcp arbitrary function of time f(t) for the time interval T determined by the formula:

Numerically mean Fav equal to the height of a rectangle equal in area to a figure bounded by a curve f(t), the axis t and the limits of integration 0 - T(fig. 35).

For a sinusoidal function, the average value over the full period T(or for an integer number of full periods) is equal to zero, since the areas of the positive and negative half-waves of this function are equal. For an alternating sinusoidal voltage, the modulus average value over the full period is determined T or the average value for half the period ( T/ 2) between two zero values ​​(Fig. 36):

Ucp = Um ∙ sin wt dt = 2R... Thus, the quantitative parameters of electrical energy on alternating current (amount of energy, power) are determined by the effective voltage values U and current I... For this reason, in the electric power industry, it is customary to perform all theoretical calculations and experimental measurements for the effective values ​​of currents and voltages. In radio engineering and communication engineering, on the contrary, they operate with the maximum values ​​of these functions.

The above formulas for AC energy and power are exactly the same as those for DC. On this basis, it can be argued that the energetically direct current is equivalent to the effective value of the alternating current.

What is taken for the effective value of the AC current and AC voltage

what is taken as the rms value of the alternating current and alternating voltage?

Battle Egg

Alternating current, in a broad sense, an electric current that changes over time. Typically, in technology, P. t. Is understood as a periodic current in which the average value over a period of current and voltage is zero.

Alternating currents and alternating voltages are constantly changing in magnitude. At every other moment they have a different magnitude. The question arises, how do you measure them? To measure them, the concept of effective value is introduced.

The effective or effective value of an alternating current is called the magnitude of such a direct current, which, in terms of its thermal effect, is equivalent to a given alternating current.

The effective or effective value of an alternating voltage is called the magnitude of such a constant voltage, which, in terms of its thermal effect, is equivalent to a given alternating voltage.

All alternating currents and voltages in technology are measured in rms values. Devices measuring variable variables show their rms value.

Question: the voltage in the electrical network is 220 V, what does this mean?

This means that a 220 V constant voltage source has the same thermal effect as the power grid.

The effective value of a sinusoidal current or voltage is 1.41 times less than the amplitude of this current or voltage.

Example: Determine the amplitude of the mains voltage with a voltage of 220 V.

The amplitude is 220 * 1.41 = 310.2 V.

additional information

In the English-language technical literature, the term “ effective value"- in literal translation" effective value»

In electrical engineering, devices of electromagnetic, electrodynamic and thermal systems react to the effective value.

Sources of

• "Handbook of Physics", Yavorskiy BM, Detlaf AA, ed. "Science", 1979 1
• Physics course. A. A. Detlaf, B. M. Yavorsky M .: Higher. shk., 1989. § 28.3, p. 5
• "Theoretical Foundations of Electrical Engineering", L. A. Bessonov: Higher. shk., 1996. § 7.8 - § 7.10

Links

see also

• List of voltage and current parameters

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See what the "RMS value of alternating current" is in other dictionaries:

r.m.s. value of alternating current

effective value of alternating current- efektinė srovė statusas T sritis Standartizacija ir metrologija apibrėžtis Apibrėžtį žr. priede. priedas (ai) Grafinis formatas atitikmenys: angl. effective current; root mean square current vok. Effektivstrom, m rus. effective value ... ... Penkiakalbis aiškinamasis metrologijos terminų žodynas

effective current- RMS value of the periodic electric current for the period. Note - RMS values ​​of periodic electric voltage, electromotive force, magnetic flux, etc. are determined similarly [GOST R 52002 2003] ... ...

In electrical engineering, the root-mean-square value of alternating current, voltage, electromotive force, magnetomotive force, magnetic flux, etc. for a period. The effective value of sinusoidal current and voltage is one times less than their amplitude ... ... Big Encyclopedic Dictionary

- (electrical), the mean square value of the alternating current, voltage, emf, magnetomotive force, magnetic flux, etc. for the period. The effective values ​​of the sinusoidal current and voltage are √2 times less than their amplitude values. * * * ... ... encyclopedic Dictionary

Wed square-law value of alternating current, voltage, emf, magnetomotive force, magn. flow, etc. D. z. sinusoidal current and voltage in sq. root of 2 times less than their amplitude values ​​... Natural science. encyclopedic Dictionary

GOST R IEC 60252-2-2008: Capacitors for AC motors. Part 2. Starting capacitors- Terminology GOST R IEC 60252 2 2008: Capacitors for AC motors. Part 2. Starting capacitors original document: 1.3.11 duty cycle duration: The total time of one loading (voltage supply) and ... ... Dictionary-reference book of terms of normative and technical documentation

true effective value Technical translator's guide

true effective value- [Intent] A device that measures a non-sinusoidal electrical signal, for example, in the form of pulses or sections of a sinusoid, taking into account all the harmonics of this signal, is a device that determines the true effective value of this signal. ... ... Technical translator's guide

true effective value- [Intent] A device that measures a non-sinusoidal electrical signal, for example, in the form of pulses or sections of a sinusoid, taking into account all the harmonics of this signal, is a device that determines the true effective value of this signal. ... ... Technical translator's guide

The physical meaning of these concepts is approximately the same as the physical meaning of the average speed or other quantities averaged over time. At different points in time, the strength of the alternating current and its voltage take on different values, therefore, talking about the strength of the alternating current in general can only be conditional.

At the same time, it is quite obvious that different currents have different energy characteristics - they produce different work in the same period of time. The work performed by the current is taken as the basis for determining the effective value of the current strength. They are set for a certain period of time and calculate the work done by alternating current during this period of time. Then, knowing this work, the reverse calculation is performed: they find out the strength of the direct current, which would produce similar work in the same period of time. That is, power averaging is performed. The calculated force of a hypothetically flowing direct current through the same conductor, producing the same work, is the effective value of the original alternating current. Do the same with tension. This calculation is reduced to determining the value of such an integral:

Where does this formula come from? From the well-known formula for the power of the current, expressed in terms of the square of its strength.

RMS values ​​of periodic and sinusoidal currents

Calculating the effective value for arbitrary currents is an unproductive exercise. But for a periodic signal, this parameter can be very useful. It is known that any periodic signal can be decomposed into a spectrum. That is, it is represented as a finite or infinite sum of sinusoidal signals. Therefore, to determine the magnitude of the effective value of such a periodic current, we need to know how to calculate the effective value of a simple sinusoidal current. As a result, by adding the RMS values ​​of the first few harmonics with the maximum amplitude, we obtain an approximate value of the RMS current value for an arbitrary periodic signal. Substituting the expression for the harmonic vibration into the above formula, we obtain the following approximate formula.